linreg - Linear Regression and Correlation Explanatory and...

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Linear Regression and Correlation Explanatory and Response Variables are Numeric Relationship between the mean of the response variable and the level of the explanatory variable assumed to be approximately linear (straight line) Model: ) , 0 ( ~ 1 0 σ ε ε β β N x Y + + = β 1 > 0 Positive Association β 1 < 0 Negative Association β 1 = 0 No Association
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Least Squares Estimation of β 0 , β 1 2200 β 0 Mean response when x =0 ( y -intercept) 2200 β 1 Change in mean response when x increases by 1 unit (slope) β 0 , β 1 are unknown parameters (like μ ) β 0 + β 1 x Mean response when explanatory variable takes on the value x Goal: Choose values (estimates) that minimize the sum of squared errors ( SSE ) of observed values to the straight-line: 2 1 1 ^ 0 ^ 1 2 ^ 1 ^ 0 ^ ^ = = + - = - = + = n i i i n i i i x y y y SSE x y β β β β
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Example - Pharmacodynamics of LSD Score (y) LSD Conc (x) 78.93 1.17 58.20 2.97 67.47 3.26 37.47 4.69 45.65 5.83 32.92 6.00 29.97 6.41 Response ( y ) - Math score (mean among 5 volunteers) Predictor ( x ) - LSD tissue concentration (mean of 5 volunteers) Raw Data and scatterplot of Score vs LSD concentration: LSD_CONC 7 6 5 4 3 2 1 SCORE 80 70 60 50 40 30 20 Source: Wagner, et al (1968)
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Least Squares Computations ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 ^ 2 1 ^ 0 ^ 2 1 ^ 2 2 - = - - = - = = - - - = - = - - = - = n SSE n y y s x y S S x x y y x x y y S y y x x S x x S xx xy yy xy xx β β β
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Example - Pharmacodynamics of LSD 72 . 50 01 . 9 10 . 89 10 . 89 ) 33 . 4 )( 01 . 9 ( 09 . 50 01 . 9 4749 . 22 4872 . 202 333 . 4 7 33 . 30 087 . 50 7 61 . 350 2 ^ 1 ^ 0 ^ 1 ^ = - = = - - = - = - = - = = = = = s x y x y x y β β β Score (y) LSD Conc (x) x-xbar y-ybar Sxx Sxy Syy 78.93 1.17 -3.163 28.843 10.004569 -91.230409 831.918649 58.20 2.97 -1.363 8.113 1.857769 -11.058019 65.820769 67.47 3.26 -1.073 17.383 1.151329 -18.651959 302.168689 37.47 4.69 0.357 -12.617 0.127449 -4.504269 159.188689 45.65 5.83 1.497 -4.437 2.241009 -6.642189 19.686969 32.92 6.00 1.667 -17.167 2.778889 -28.617389 294.705889 29.97 6.41 2.077 -20.117 4.313929 -41.783009 404.693689 350.61 30.33 -0.001 0.001 22.474943 -202.487243 2078.183343 (Column totals given in bottom row of table)
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SPSS Output and Plot of Equation Coefficients a 89.124 7.048 12.646 .000 -9.009 1.503 -.937 -5.994 .002 (Constant) LSD_CONC Model 1 B Std. Error Unstandardized Coefficients Beta Standardized Coefficients t Sig. Dependent Variable: SCORE a. Linear Regression 1.00 2.00 3.00 4.00 5.00 6.00 lsd_conc 30.00 40.00 50.00 60.00 70.00 80.00 score c c c c c c c score = 89.12 + -9.01 * lsd_conc R-Square = 0.88 Math Score vs LSD Concentration (SPSS)
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Inference Concerning the Slope ( β 1 ) Parameter: Slope in the population model ( β 1 ) Estimator: Least squares estimate: Estimated standard error: Methods of making inference regarding population: Hypothesis tests (2-sided or 1-sided) Confidence Intervals 1 ^ β xx S s / ^ 1 ^ = β σ
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Hypothesis Test for β 1 2-Sided Test H 0 : β 1 = 0 H A : β 1 0 1-sided Test H 0 : β 1 = 0 H A + : β 1 > 0 or H A - : β 1 < 0 |) | ( 2 : | | : . . : . . 2 , 2 / ^ 1 ^ 1 ^ obs n obs obs t t P val P t t R R t S T - = - α β σ β ) ( : ) ( : : . . : . .
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